I will talk about "complex cells", a holomorphic counterpart to the standard cellular decompositions used in o-minimal geometry, that we defined in a joint work with Novikov. Our initial motivation was a conjecture of Yomdin's concerning tail entropies of real-analytic maps, and some applications around diophantine geometry. In retrospect we realized that complex cells are also closely related to the subanalytic preparation theorems by Parusinski and Lion-Rolin. I'll explain how ideas from hyperbolic geometry shed new light on these theorems. I'll finish with some partly conjectural outline of how this notion could improve our understanding of the effectivity of various classical constructions in o-minimality such as Wilkie's theorem of the complement.